{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:GMYQKB5OELC5UTE26JFXDH4XJY","short_pith_number":"pith:GMYQKB5O","canonical_record":{"source":{"id":"2506.00349","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-31T02:17:39Z","cross_cats_sorted":[],"title_canon_sha256":"0e0741ebc4806799f9645a593f16414fdad6c58702e9201ab467255ff562aa18","abstract_canon_sha256":"d6f21d6d4b9dca790dda8653184cc601ac19f085d354d0aa2a0e33aa432cdd08"},"schema_version":"1.0"},"canonical_sha256":"33310507ae22c5da4c9af24b719f974e1d1f02fe511572936d6bfcffceccdece","source":{"kind":"arxiv","id":"2506.00349","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.00349","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"arxiv_version","alias_value":"2506.00349v1","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.00349","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"pith_short_12","alias_value":"GMYQKB5OELC5","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"pith_short_16","alias_value":"GMYQKB5OELC5UTE2","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"pith_short_8","alias_value":"GMYQKB5O","created_at":"2026-07-05T11:13:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:GMYQKB5OELC5UTE26JFXDH4XJY","target":"record","payload":{"canonical_record":{"source":{"id":"2506.00349","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-31T02:17:39Z","cross_cats_sorted":[],"title_canon_sha256":"0e0741ebc4806799f9645a593f16414fdad6c58702e9201ab467255ff562aa18","abstract_canon_sha256":"d6f21d6d4b9dca790dda8653184cc601ac19f085d354d0aa2a0e33aa432cdd08"},"schema_version":"1.0"},"canonical_sha256":"33310507ae22c5da4c9af24b719f974e1d1f02fe511572936d6bfcffceccdece","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:13:33.037580Z","signature_b64":"E66w8wLhksHtGRBXiIqXjsGI3yHqsh2Gy6kARc3O6f++qutewJc2piEW3PBN5ebZB91OWnyUotRi2eL7tQ9qBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"33310507ae22c5da4c9af24b719f974e1d1f02fe511572936d6bfcffceccdece","last_reissued_at":"2026-07-05T11:13:33.037154Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:13:33.037154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.00349","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:13:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yZcP/nh0vR5QPYM8iQPECEv40sWmN9Ciz3RQrA6cvraol2o6cPJRHN6J6j1XR9/9q7ynAmlb5cThuxo1SfmcBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:29:37.428137Z"},"content_sha256":"5a4c34786c62508f036560305229024ca04406011f34c70758aca4900e1ed2cd","schema_version":"1.0","event_id":"sha256:5a4c34786c62508f036560305229024ca04406011f34c70758aca4900e1ed2cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:GMYQKB5OELC5UTE26JFXDH4XJY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chau Nguyen, Dora Woodruff, Son Nguyen","submitted_at":"2025-05-31T02:17:39Z","abstract_excerpt":"We give a new formula for the Littlewood--Richardson coefficients in terms of peelable tableaux compatible with shuffle tableaux, in the same fashion as Remmel--Whitney rule. This gives an efficient way to compute generalized Littlewood--Richardson coefficients for Temperley--Lieb immanants of Jacobi--Trudi matrices. We will also show that our rule behaves well with Bender--Knuth involutions, recovering the symmetry of Littlewood--Richardson coefficients. As an application, we use our rule to prove a special case of a Schur log-concavity conjecture by Lam--Postnikov--Pylyavskyy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.00349","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.00349/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:13:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u3GXJ6KPUTOfmk3Aet/CAiU3IsEQXMcQkpAgSAqb6M1yCToehwJLmHr27wuz3yaeHHSJ722ns8SAhwVXkmsLDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T08:29:37.428513Z"},"content_sha256":"c8d0cc7f6de2fafa563727e1ceb7205bf5fc450d7a741ed2fcd59dad46491bdd","schema_version":"1.0","event_id":"sha256:c8d0cc7f6de2fafa563727e1ceb7205bf5fc450d7a741ed2fcd59dad46491bdd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GMYQKB5OELC5UTE26JFXDH4XJY/bundle.json","state_url":"https://pith.science/pith/GMYQKB5OELC5UTE26JFXDH4XJY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GMYQKB5OELC5UTE26JFXDH4XJY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-07T08:29:37Z","links":{"resolver":"https://pith.science/pith/GMYQKB5OELC5UTE26JFXDH4XJY","bundle":"https://pith.science/pith/GMYQKB5OELC5UTE26JFXDH4XJY/bundle.json","state":"https://pith.science/pith/GMYQKB5OELC5UTE26JFXDH4XJY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GMYQKB5OELC5UTE26JFXDH4XJY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:GMYQKB5OELC5UTE26JFXDH4XJY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d6f21d6d4b9dca790dda8653184cc601ac19f085d354d0aa2a0e33aa432cdd08","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-31T02:17:39Z","title_canon_sha256":"0e0741ebc4806799f9645a593f16414fdad6c58702e9201ab467255ff562aa18"},"schema_version":"1.0","source":{"id":"2506.00349","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.00349","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"arxiv_version","alias_value":"2506.00349v1","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.00349","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"pith_short_12","alias_value":"GMYQKB5OELC5","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"pith_short_16","alias_value":"GMYQKB5OELC5UTE2","created_at":"2026-07-05T11:13:33Z"},{"alias_kind":"pith_short_8","alias_value":"GMYQKB5O","created_at":"2026-07-05T11:13:33Z"}],"graph_snapshots":[{"event_id":"sha256:c8d0cc7f6de2fafa563727e1ceb7205bf5fc450d7a741ed2fcd59dad46491bdd","target":"graph","created_at":"2026-07-05T11:13:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2506.00349/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We give a new formula for the Littlewood--Richardson coefficients in terms of peelable tableaux compatible with shuffle tableaux, in the same fashion as Remmel--Whitney rule. This gives an efficient way to compute generalized Littlewood--Richardson coefficients for Temperley--Lieb immanants of Jacobi--Trudi matrices. We will also show that our rule behaves well with Bender--Knuth involutions, recovering the symmetry of Littlewood--Richardson coefficients. As an application, we use our rule to prove a special case of a Schur log-concavity conjecture by Lam--Postnikov--Pylyavskyy.","authors_text":"Chau Nguyen, Dora Woodruff, Son Nguyen","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-31T02:17:39Z","title":"Shuffle Tableaux, Littlewood--Richardson Coefficients, and Schur Log-Concavity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.00349","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:5a4c34786c62508f036560305229024ca04406011f34c70758aca4900e1ed2cd","target":"record","created_at":"2026-07-05T11:13:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d6f21d6d4b9dca790dda8653184cc601ac19f085d354d0aa2a0e33aa432cdd08","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-05-31T02:17:39Z","title_canon_sha256":"0e0741ebc4806799f9645a593f16414fdad6c58702e9201ab467255ff562aa18"},"schema_version":"1.0","source":{"id":"2506.00349","kind":"arxiv","version":1}},"canonical_sha256":"33310507ae22c5da4c9af24b719f974e1d1f02fe511572936d6bfcffceccdece","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"33310507ae22c5da4c9af24b719f974e1d1f02fe511572936d6bfcffceccdece","first_computed_at":"2026-07-05T11:13:33.037154Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:13:33.037154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E66w8wLhksHtGRBXiIqXjsGI3yHqsh2Gy6kARc3O6f++qutewJc2piEW3PBN5ebZB91OWnyUotRi2eL7tQ9qBg==","signature_status":"signed_v1","signed_at":"2026-07-05T11:13:33.037580Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.00349","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5a4c34786c62508f036560305229024ca04406011f34c70758aca4900e1ed2cd","sha256:c8d0cc7f6de2fafa563727e1ceb7205bf5fc450d7a741ed2fcd59dad46491bdd"],"state_sha256":"469ec36164db66f155dcd95e8abcb11d53b85f9e548ad7f79b723bb37f49dd42"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jeyZC0r1Y79eubKkg2k6ImRIQBtm9mct+rb708RVp4n1PIN/jrRY7bUP836hEfQvo9Gc+fvimTKG1itWIgpkAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T08:29:37.430514Z","bundle_sha256":"670b69215b89b5715709d2819a888930631fa56a27e1f593da9e17d0e5774cf5"}}